the 3rd dimension ?

[tommy1729]

till now we considered tetration for 2D complex numbers.

what about 3D numbers ?

are there methods for 3D numbers that are distinct from the 2D ?

the problem might be that the Riemann Mapping Theorem does not apply in 3D. (only conformal mapping are moebius in 3D)

i think there is no 3D solution and we will need to use the 2D solutions and apply them to get a 3D solution with is correct upto its " complex absolute value ".

it should be noted that there are 2 types of 3D numbers.

a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.

( group ring is the correct term here )

and the classical " 3D complex "

a + b w + c w^2 where a , b , c are real and w^3 = 1

( it is trivial to compute the " absolute complex value " , just replace P with i or w with the upper cube root of unity )

the advantage in 3D might be less fixpoints for exp^[r](z) = exp(z) and cycle detection / branch point understanding of the ordinary 2D sexp / slog.

( this relates to some threads like tid616 and tid499 amongst others )

so let me know what you think.

regards

tommy1729

[bo198214]

I never heard about 3 dimensional numbers, the possible finite dimensional division algebras must have dimension 1 (real), 2 (complex), 4 (quaternion) or 8 (octonion), see wikipedia.
I remember that Hamilton tried to find to 3 dimensional numbers but failed, and came up in the end with quaternions.

it should be noted that there are 2 types of 3D numbers.

a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.

Isnt that 4 dimensional, a,b,c,d?

and the classical " 3D complex "

a + b w + c w^2 where a , b , c are real and w^3 = 1

How do you define division here?

[tommy1729]

I never heard about 3 dimensional numbers, the possible finite dimensional division algebras must have dimension 1 (real), 2 (complex), 4 (quaternion) or 8 (octonion), see wikipedia.
I remember that Hamilton tried to find to 3 dimensional numbers but failed, and came up in the end with quaternions.

yes and no : we have zero-divisors in 3D.

it should be noted that there are 2 types of 3D numbers.

a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.

Isnt that 4 dimensional, a,b,c,d?

no , since like i said : a , b , c , d are POSITIVE.

the units are not orthogonal.

and the classical " 3D complex "

a + b w + c w^2 where a , b , c are real and w^3 = 1

How do you define division here?

just as the multiplicative inverse.

1 = (a' + b' w + c' w^2)(a + b w + c w^2)

if (a + b w + c w^2) is not a zero-divisor.

---

in abstract algebra notation the 2 kinds of 3D numbers are RxRxR and RxC.

matrix representation is a must , and they also satisfy the modified Cauchy-Riemann equations.

( they also extend the gaussian integers , which makes fun number theory but that is a bit off topic , also its possible to use them for rotations rather than euler angles and quaternions )

perhaps the following are illuminating :

http://en.wikipedia.org/wiki/Group_ring

( amateur mathematician ) http://bandtech.com/PolySigned/PolySigned.html
( also check the links )

many papers by beresford or Silviu Olariu ( cant find them right now )

http://en.wikipedia.org/wiki/Tricomplex_number

http://arxiv.org/PS_cache/math/pdf/0008/0008120v1.pdf

http://arxiv.org/PS_cache/math/pdf/0011/0011044v1.pdf

and many others.


feel free to give more free references !

regards

tommy1729

[bo198214]

Then go ahead, give us a taste how 3-complex numbers could have benefits for tetration. I guess these numbers are completely new for most of the forum members.
(I always wonder why you are hiding most of the information that you seem to know in your posts)

[tommy1729]

i was just toying with the idea.